By default (and rightly so), CORS is disabled.
You will have to use JSON-P to circumvent this. (Twitch docs)
The default way to do it with jQuery is setting callback=? in the url, and it will automatically use some JSON-P magic. (jQuery Docs)
EDIT: Good page on CORS: https://developer.mozilla.org/en-US/docs/Web/HTTP/Access_control_CORS